Local Gravity versus Local Velocity: Solutions for $\beta$ and nonlinear bias

(abridged) We perform a reconstruction of the cosmological large scale flows in the nearby Universe using two complementary observational sets. The first, the SFI++ sample of Tully-Fisher (TF) measurements of galaxies, provides a direct probe of the flows. The second, the whole sky distribution of galaxies in the 2MASS redshift survey (2MRS), yields a prediction of the flows given the cosmological density parameter, $\Omega$, and a biasing relation between mass and galaxies. We aim at an unbiased comparison between the peculiar velocity fields extracted from the two data sets and its implication on the cosmological parameters and the biasing relation. We expand the fields in a set of orthonormal basis functions, each representing a plausible realization of a cosmological velocity field. Our analysis completely avoids the strong error covariance in the smoothed TF velocities by the use of orthonormal basis functions and employs elaborate realistic mock data sets to extensively calibrate the errors in 2MRS predicted velocities. We relate the 2MRS galaxy distribution to the mass density field by a linear bias factor, $b$, and include a luminosity dependent, $\propto L^\alpha$, galaxy weighting. We assess the agreement between the fields as a function of $\alpha$ and $\beta=f(\Omega)/b$, where $f$ is the growth factor of linear perturbations. The agreement is excellent with a reasonable $\chi^2$ per degree of freedom. For $\alpha=0$, we derive $0.28<\beta<0.37$ and $0.24<\beta<0.43$, respectively, at the 68.3% and 95.4% confidence levels (CLs). For $\beta=0.33$, we get $\alpha<0.25$ and $\alpha<0.5$, respectively, at the 68.3% and 95.4% CLs. We set a constraint on the fluctuation normalization, finding $\sigma_8 = 0.73 \pm 0.1$, in very good agreement with the latest WMAP results.